with \(\mathbf {x}[0] = \mathbf {x}_{0} \in \mathbb {X}_{0}\)  is locally asymptotically stable in \(\mathbb {X}_{0}\)  without violating any constraints in the original MPC problem (REF ).



The proof of the theorem is a direct application of Theorem 13.1 in [1]}.

Note that inequality conditions (REF ) are typically verified through sampling and interpolation [1]}, which makes this approach only suitable for relatively simple, moderately sized problems.
Besides, the above condition is only a sufficient condition, for which we can safely discard the fail-safe MPC mode.
Other methods are possible.
For example, [3]}, [4]} have proposed methods to verify the stability of a NN controller using Lyapunov approaches.
